How Do You Solve a PDE Involving Heat Transfer in a Circular Heat Source?

[mehsus]

I have a circular heat source of inner radius r1 and r2=r1+Δr on top of a pcb board. This heat source is transferring heat along the radius and the length of the beaker which is say L. I have to find temperature distribution along the length of the beaker so T(r.z). The beaker is filled with water till the bottom of the pcb.

I considered a differential element of radius Δr and thickness Δz at a distance z from the bottom of the pcb. Now once I write the energy balance equation on the basis of steady state as:

Q_z-Q_z+Δz+Q_r-Q_r+Δr=0

There is no convection taking place.

I come to the PDE:

d^2T/dz^2+1/r(dt/dr)+d^2T/dr^2=0

I am stuck here in solving this PDE...please help

 

[fzero]

Have you tried separation of variables? I.e., try to find solutions of the form $T(r,z) = R(r) Z(z)$.

 

[HallsofIvy]

It looks to me that after you separate variables, you will get Bessel's equation in r.